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Quantum tomography under prior information

Teiko Heinosaari, Luca Mazzarella, Michael, M. Wolf

Published Aug 17, 2016• 160 citations
OpenAlex

Abstract

Abstract. We provide a detailed analysis of the question: how many measurement settings or outcomes are needed in order to identify a quantum system which is constrained by prior information? We show that if the prior information restricts the system to a set of lower dimensionality, then topological obstructions can increase the required number of outcomes by a factor of two over the number of real parameters needed to characterize the system. Conversely, we show that almost every measurement becomes informationally complete with respect to the constrained set if the number of outcomes exceeds twice the Minkowski dimension of the set. We apply the obtained results to determine the minimal number of outcomes of measurements which are informationally complete with respect to states with rank constraints. In particular, we show that 4d − 4 measurement outcomes (POVM elements) is enough in order to identify all pure states in a d-dimensional Hilbert space, and that the minimal number is at most 2 log2(d) smaller than this upper bound. 1.

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arXiv:1308.5502•Journal of Physics A Mathematical and Theoretical•Feb 3, 2014•24 citations
Abstract. We provide a detailed analysis of the question: how many measurement settings or outcomes are needed in order to identify a quantum system which is constrained by prior information? We show that if the prior information restricts the system to a set of lower dimensionality, then topological obstructions can increase the required number of outcomes by a factor of two over the number of real parameters needed to characterize the system. Conversely, we show that almost every measurement becomes informationally complete with respect to the constrained set if the number of outcomes exceeds twice the Minkowski dimension of the set. We apply the obtained results to determine the minimal number of outcomes of measurements which are informationally complete with respect to states with rank constraints. In particular, we show that 4d − 4 measurement outcomes (POVM elements) is enough in order to identify all pure states in a d-dimensional Hilbert space, and that the minimal number is at most 2 log2(d) smaller than this upper bound. 1.
This dissertation is concerned with quantum state tomography in scenarios where prior information constrains the set of relevant quantum states to a low dimensional subset of the state space. Frameworks to find both lower and upper bounds on the minimal number of measurement settings or outcomes needed to discriminate any two quantum states of such a subset are provided and applied to several concrete scenarios. Furthermore, measurement settings that allow a computationally tractable recovery of bounded rank quantum states are constructed.
We investigate quantum tomography in scenarios where prior information restricts the state space to a smooth manifold of lower dimensionality. By considering stability we provide a general framework that relates the topology of the manifold to the minimal number of binary measurement settings that is necessary to discriminate any two states on the manifold. We apply these findings to cases where the subset of states under consideration is given by states with bounded rank, fixed spectrum, given unitary symmetry or taken from a unitary orbit. For all these cases we provide both upper and lower bounds on the minimal number of binary measurement settings necessary to discriminate any two states of these subsets.
We analyze quantum state tomography in scenarios where measurements and states are both constrained. States are assumed to live in a semi-algebraic subset of state space, and measurements are supposed to be rank-one positive operator-valued measures, possibly with additional constraints. Specifically, we consider sets of von Neumann measurements and sets of local observables. We provide upper bounds on the minimal number of measurement settings or outcomes that are required for discriminating all states within the given set. The bounds exploit tools from real-algebraic geometry and lead to generic results that do not only show the existence of good measurements, but guarantee that almost all measurements with the same dimension characteristic perform equally well. In particular, we show that on an |nnn|-dimensional Hilbert space any two states of a semi-algebraic subset can be discriminated by |kkk| generic von Neumann measurements if |k(n−1)k(n-1)k(n−1)| is larger than twice the dimension of the subset. In case the subset is given by states of rank at most |rrr|⁠, we show that |kkk| generic von Neumann measurements suffice to discriminate any two states provided that |k(n−1)>4r(n−r)−2k(n-1)>4r(n-r)-2k(n−1)>4r(n−r)−2|⁠. We obtain corresponding results for low-rank matrix recovery of hermitian matrices in the scenario where the linear measurement mapping is induced by tight frames.
Quantum state tomography seeks to reconstruct an unknown state from measurement statistics. A finite measurement (POVM) is \emph{pure-state informationally complete} (PSI-Complete) if the outcome probabilities determine any pure state up to a global phase. We study \emph{rank-one} POVMs that are minimally sufficient for this task. We call such a POVM \emph{vital} if it is PSI-Complete but every proper subcollection is not PSI-Complete. We prove sharp upper bounds on the size of vital rank-one POVMs in dimension nnn: the size is at most (n+12)\binom{n+1}{2}(2n+1​) over R\mathbb{R}R and at most n2n^{2}n2 over C\mathbb{C}C, and we give constructions that attain these bounds. In the real case, we further exhibit a connection to block designs: whenever w∣n(n−1)w \mid n(n-1)w∣n(n−1), an (n,w,w−1)(n,w,w-1)(n,w,w−1) design produces a vital rank-one POVM with n+n(n−1)/wn + n(n-1)/wn+n(n−1)/w outcomes. We provide explicit constructions for w=2,n−1w=2,n-1w=2,n−1, and nnn.
We study quantum process tomography given the prior information that the map is a unitary or close to a unitary process. We show that a unitary map on a ddd-level system is completely characterized by a minimal set of d2+d{d}^{2}+dd2+d elements associated with a collection of POVMs, in contrast to the ${d}^{4}\ensuremath{-}{d}^{2}$ elements required for a general completely positive trace-preserving map. To achieve this lower bound, one must probe the map with particular sets of ddd pure states. We further compare the performance of different estimators inspired by compressed sensing, to reconstruct a near-unitary process from such data. We find that when we have accurate prior information, an appropriate compressed sensing method reduces the required data needed for high-fidelity estimation, and different estimators applied to the same data are sensitive to different types of noise. Convex optimization inspired by the techniques of compressed sensing can therefore be used both as indicators of error models and to validate the use of the prior assumptions.
The purpose of quantum tomography is to determine an unknown quantum state from measurement outcome statistics. There are two obvious ways to generalize this setting. First, our task need not be the determination of any possible input state but only some input states, for instance pure states. Second, we may have some prior information, or premise, which guarantees that the input state belongs to some subset of states, for instance the set of states with rank less than half of the dimension of the Hilbert space. We investigate state determination under these two supplemental features, concentrating on the cases where the task and the premise are statements about the rank of the unknown state. We characterize the structure of quantum observables (positive operator valued measures) that are capable of fulfilling these type of determination tasks. After the general treatment we focus on the class of covariant phase space observables, thus providing physically relevant examples of observables both capable and incapable of performing these tasks. In this context, the effect of noise is discussed.